from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1032, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,21,10]))
pari: [g,chi] = znchar(Mod(53,1032))
Basic properties
| Modulus: | \(1032\) | |
| Conductor: | \(1032\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1032.cf
\(\chi_{1032}(53,\cdot)\) \(\chi_{1032}(101,\cdot)\) \(\chi_{1032}(197,\cdot)\) \(\chi_{1032}(341,\cdot)\) \(\chi_{1032}(461,\cdot)\) \(\chi_{1032}(533,\cdot)\) \(\chi_{1032}(701,\cdot)\) \(\chi_{1032}(797,\cdot)\) \(\chi_{1032}(869,\cdot)\) \(\chi_{1032}(917,\cdot)\) \(\chi_{1032}(941,\cdot)\) \(\chi_{1032}(1013,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((775,517,689,433)\) → \((1,-1,-1,e\left(\frac{5}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1032 }(53, a) \) | \(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) |
sage: chi.jacobi_sum(n)